Let P(x) be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers n such that P(n) is composite.
Hint(s):
Remember that if a and b are distinct integers, then P(a) - P(b) is divisible by a - b.
+876 The simplest polynomial is ax2+bx+c
Let x=k⋅c for k∈Z
We have ax2+bx+c=a(kc)2+b(kc)+c=ak2c2+bkc+c=c(ak2c+bk+1)
Because there is are at least two factors for ax2+bx+c, P(x) is composite.
